Stochastic complement analysis of multi-server threshold queues with hysteresis

We consider a K-server threshold-based queueing system with hysteresis, for which a set of forward thresholds (F1,F2,…,FK−1) and a set of reverse thresholds (R1,R2,…,RK−1) are defined. A simple version of this multi-server queueing system behaves as follows. When a customer arrives to an empty system, it is serviced by a single server. Whenever the number of customers exceeds a forward threshold Fi, a server is added to the system and server activation is instantaneous. Whenever the number of customer falls below a reverse threshold Ri, a server is removed from the system. We consider and solve several variations of this problem, namely: (1) homogeneous servers with Poisson arrivals, (2) homogeneous servers with bulk (Poisson) arrivals, and (3) heterogeneous servers with Poisson arrivals. We place no restrictions on the number of servers or the bulk sizes or the size of the waiting room. In [O.C. Ibe, J. Keilson, Multi-server threshold queues with hysteresis, Performance Evaluation 21 (1995) 185–212], the authors solve a limited form of this problem using Green’s function method. More specifically, they give a closed-form solution for a K-server system, when the servers are homogeneous, and for a two-server system, when the servers are heterogeneous; the authors experienced difficulties in extending Green’s function method beyond the case of two heterogeneous servers. Rather than using Green’s function, we solve this problem using the concept of stochastic complementation, which is a more intuitive and more easily extensible method. For the case of a homogeneous multi-server system we are able to derive a closed-form solution for the steady state probability vector; for the remaining cases we give an algorithmic solution. Note, however, that we can use stochastic complementation to derive closed-form solutions for some limited forms of cases (2) and (3), such as heterogeneous servers with K=2 and bulk arrivals with a limited bulk size. Finally, our technique works both for systems with finite and infinite waiting rooms.